BRIEF NOTEJapanese Journal of Applied Physics 58, 028003 (2019)https://doi.org/10.7567/1347-4065/aaf897

Lorentzian demodulation algorithm for multimode polymer optical fiber Bragg
gratings
Yosuke Mizuno1* , Tianyi Ma1, Ryo Ishikawa1, Heeyoung Lee1 , Antreas Theodosiou2 , Kyriacos Kalli2 , andKentaro Nakamura1
1

Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8503, Japan
Photonics and Optical Sensing Research Laboratory, Cyprus University of Technology, Limassol 3036, Cyprus

2*E-mail: ymizuno@sonic.pi.titech.ac.jp

Received November 14, 2018; accepted December 17, 2018; published online January 10, 2019
For strain sensing applications, we develop a simple and accurate Lorentzian demodulation algorithm for fiber Bragg gratings (FBGs) inscribed in
multimode optical fibers. As multimode FBG spectra generally exhibit multiple peaks, simply tracking some particular spectral peaks (i.†e.,
maximum detection) results in measurement failure when the strain is large. Here, using an FBG inscribed in a perfluorinated graded-index
multimode polymer optical fiber, we experimentally evaluate the new method by comparing its performance with those of other demodulation
algorithms, such as the maximum and centroid detection methods. Finally, using this method, we demonstrate the accurate measurement of strainof up to 1.0%. © 2019 The Japan Society of Applied Physics

Fiber Bragg gratings (FBGs) have been extensively studied
because of their capability to highly accurate measure various
physical parameters including strain, temperature, andhumidity.1–8) Recently, FBGs have been inscribed not onlyin single-mode ﬁbers (SMFs) but also in multimode ﬁbers(MMFs).9–11) As MMF-FBG reﬂection spectra generally
exhibit multiple peaks corresponding to different propagation
modes, an accurate demodulation algorithm for detecting the
Bragg wavelengths from multiple-peak FBG spectra is highly
desirable.
MMFs in which FBGs can be inscribed include not only
glass optical ﬁbers but also polymer optical ﬁbers (POFs).12)
Owing to their high ﬂexibility, POF-FBGs have been
exploited for unique sensing applications.13–15) As well as
in standard POFs composed of polymethyl methacrylate,
FBGs have been successfully inscribed in perﬂuorinatedgraded-index (PFGI) POFs16) with relatively low loss attelecom wavelength by the use of a femtosecond laser.17,18)
Although all the commercially available PFGI-POFs are
MMFs, by exploiting some demodulation algorithms, they
have already been used to develop various kinds ofsensors.19–21) Note that it is difﬁcult and not practicallyconvenient to excite a single peak (corresponding to thefundamental mode) in the PFGI-POF-FBGs.22)
Of all the demodulation algorithms previously reported,
what is simplest is the maximum detection,19,23) i.e., tracking
a maximal peak (or some particular peaks). However, thismethod results in measurement failure when the strain (ortemperature change, etc) is large, because, with increasingstrain, the multiple peaks overlap with each other or get
separated, as different peaks have different sensitivities. A
centroid detection,23) which is not largely inﬂuenced by such
distortions of a single peak, has also been used, but it islargely affected by the background noise and the wavelength
range used for demodulation. Cross-correlation,23,24) which isalso free from the inﬂuence of single-peak distortions, is agood way to solve these issue, but is reported to suffer from aso-called peak-locking effect.25,26) To suppress this drawback, a large amount of calculation time is required.27)
Recently, to further mitigate this shortcoming of the crosscorrelation method, Ref. 28 developed an accurate, highspeed method by the combined use of cross-correlation and
Hilbert transformation. Although this method can performaccurate detection of the Bragg wavelength with a short
calculation time, the calculation process is relatively complicated and it has been applied only to a relatively small strainof less than 0.1%.
In this work, we develop a simple and accurate demodulation algorithm based on Lorentzian approximation for
detecting the Bragg wavelength of the multiple-peak FBG
spectrum. Using a PFGI-POF-FBG sample, ﬁrst, we showthat the maximum and centroid detection methods lead to
non-ideal results with insufﬁcient accuracy. Then, we demonstrate the new demodulation method and prove its
effectiveness by measuring a relatively large strain of up to
1.0%.
As mentioned above, FBGs can be inscribed not only in
SMFs but also in MMFs. Suppose a GI-MMF with arefractive index proﬁle expressed as (0 < r < R):⎛ r ⎞gn (r ) = n1 1 - 2D ⎜ ⎟ ,
⎝ R⎠

(1 )

n (r ) = n1 1 - 2D ,(2 )and (R < r):where n1 is the effective refractive index at the FBG
reﬂection band, R is the core radius, Δ is the relative index
difference between core and cladding, and g is the parabolic
proﬁle parameter. Then, the approximate number M of modesin the GI-MMF is given by:29)⎛ 2pa ⎞2
g
M=⎜
n⎟D
,
⎝ l 1⎠ g + 2(3 )where λ is the FBG reﬂection wavelength. If we substitutethe parameters of a standard PFGI-POF (n1 = 1.347,
a = 25 um, Δ = ∼0.01, g = 2, λ = ∼1560 nm), the numberof modes is estimated to be around 90 at this wavelength.
When light is injected into this PFGI-POF (without any
special control for exciting only the fundamental mode),22)
some of these modes (especially, lower-order modes) are
excited, generating multiple peaks at slightly different
wavelengths in the FBG-reﬂected spectrum.
To detect the Bragg wavelength shift of the multiplepeak PFGI-POF-FBG spectrum, we propose a Lorentzian

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Jpn. J. Appl. Phys. 58, 028003 (2019)

Y. Mizuno et al.

Fig. 1. (Color online) Schematic calculation procedures of the new method for detecting the Bragg wavelength. (a) FBG-reﬂected spectrum measured with ahigh resolution (step 0). (b) Spectrum intentionally measured with an extremely low resolution (step 1; optional). (c) Spectrum with the noise ﬂoor subtracted(step 2). (d) Lorentzian-ﬁtted spectrum, from which the Bragg wavelength is derived (step 3).

demodulation algorithm. The calculation procedures areschematically depicted in Figs. 1(a)–1(d). Figure 1(a) shows
an example of the target FBG-reﬂected spectrum withmultiple peaks on a logarithm scale. As a ﬁrst step, if
necessary, this spectrum can be obtained using an optical
spectrum analyzer (OSA) with a low frequency resolution[Fig. 1(b)]. This is an optional step, which is sometimes
beneﬁcial when we use a low-cost OSA with a frequency
resolution insufﬁcient for accurate wavelength measurementsof a single peak corresponding to a certain propagation mode.
As a second step, we subtract the spectral noise ﬂoor of alight source from the measured spectrum [Fig. 1(c)]. As athird step, we ﬁt the spectrum with a Lorentzian curve:A
L (l ) =,⎛ l - l o ⎞2
⎟
1+⎜
⎝ W ⎠

(4 )

where A is the power of the spectral peak, λ0 is the central
wavelength, and W is the FWHM, and derive its λ0 value as
the Bragg wavelength [Fig. 1(d)]. These procedures are freefrom time-consuming and/or complicated calculations, andthe sampling rate of the Bragg wavelength is solely determined by the sweeping frequency of the OSA. As experimentally shown later in this paper, this algorithm is almost
free from the inﬂuence by distortions of some particular
spectral peaks (as in the maximum detection) or by thespectral shape of the background noise and the wavelength
range used for calculation (as in the centroid detection).
Besides, this algorithm is not inﬂuenced by the peak-locking
effect25,26) when the spectral position shifts in wavelength byless than the sampling resolution of the OSA.
A 2-mm-long FBG was inscribed in a 1.4-m-long
PFGI-POF.16) The PFGI-POF (GigaPOF-50SR, Chromis
Fiberoptics) has a three-layered structure: core (diameter:
50 μm; refractive index: ∼1.35), cladding (diameter: 70 μm),
and overcladding (diameter: 490 μm). The core and cladding
are composed of doped and undoped amorphous ﬂuoropolymer (CYTOP®), respectively, and the overcladding ofpolycarbonate. The optical propagation loss is relatively low(∼0.25 dB m−1) even at 1550 nm. The FBG was inscribed tothe midpoint of the PFGI-POF without removal of theovercladding using a femtosecond laser system (High Q
femtoREGEN) operating at 517 nm, with a pulse duration of
220 fs, a repetition rate of 1 kHz, and laser pulse energy of∼100 nJ. The PFGI-POF was mounted on an air bearing
translation system for accurate two-axis motion and the laser

Fig. 2. (Color online) Schematic of the experimental setup. ASE: ampliﬁed
spontaneous emission.

beam was focused from above using a long working distance×50 objective. Accurate synchronization of the laser pulse
repetition rate and stage motion allowed for plane-by-planegrating inscription, writing the grating to the desired lengthand index modulation value.
Figure 2 shows the experimental setup for observing thespectrum of the light reﬂected from the FBG inscribed in the
PFGI-POF. The output from an ampliﬁed spontaneousemission (ASE) source was injected into the PFGI-POF,
and the reﬂected light from the FBG was guided via an
optical circulator to an OSA (AQ6370, Yokogawa). A 0.2-mlong section of the PFGI-POF including the FBG wasstrained using automatic translation stages. The room temperature was 25 °C.
Figure 3(a) shows the measured strain dependence of the
FBG-reﬂected spectrum (corresponding to the 4th gratingorder)21) when the resolution of the OSA was 0.02 nm(highest). Multiple peaks corresponding to different modes
were observed. With increasing strain, the Bragg wavelength
shifted to a longer wavelength. To quantitatively evaluate thestrain dependence of the Bragg wavelength, some demodulation techniques need to be employed.
Figure 3(b) shows the strain dependence of the Bragg
wavelength when the maximum detection method was used,
i.†e., when the central wavelength of the spectral peak withthe highest power was tracked. The dependence was almost
linear, but the Bragg wavelength hopping was observed
4 times in this strain range. Consequently, four different
coefﬁcients were obtained depending on the strain range, oneof them (∼12.6 nm/%) was ∼12% different from thepreviously reported correct value (∼14.3 nm/%).13) This
result originated from the overlap and/or separation of thespectral peaks when strain was applied. As the Bragg
wavelength and the strain are not in one-to-one

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Y. Mizuno et al.

Fig. 3. (Color online) (a) Strain dependence of the FBG-reﬂected spectrum. (b) Strain dependence of the Bragg wavelength calculated using the maximum
detection method. (c) Strain dependence of the Bragg wavelength calculated using the centroid detection method.

Fig. 4. (Color online) (a) FBG-reﬂected spectrum (when strain was 0.50%) measured with the lowest resolution (2.00 nm) on the optical spectrum analyzer.
(b) FBG-reﬂected spectrum from which the noise ﬂoor was subtracted. (c) Lorentzian-ﬁtted FBG-reﬂected spectrum.

correspondence in this situation, the maximum detection
method was conﬁrmed not to be directly available for strain
sensing.
Figure 3(c) shows the strain dependence of the Bragg
wavelength when the centroid detection method was used.
When the centroids were calculated, the unit of the verticalaxis was converted from dBm (log) into mW (linear). The
spectral data in the wavelength range from 1550 to 1580 nm
were used to calculate the centroids. The calculated Bragg
wavelength dependence was far from being linear and was
not even in one-to-one correspondence to strain, whichindicates that the centroid detection method cannot bedirectly used for strain sensing. The nonlinear and nonmonotonic dependence was clearly caused by the nonnegligible noise ﬂoor. Namely, when the strain was larger,
the power of the FBG-reﬂected peaks became comparable tothe noise ﬂoor, the contribution of which to the centroid
became signiﬁcant. This also explains the reason why thestrain-induced wavelength shift of the centroid (<5 nm) wasmuch smaller than that of the spectral peak (>13 nm). Thus,
the centroid detection method was conﬁrmed to be unsuitable
for strain sensing unless the noise ﬂoor is negligibly ﬂat. It isalso clear that, with such a considerable noise ﬂoor, the result
will be largely affected by the wavelength range used for
calculation.

Subsequently, Figs. 4(a)–4(c) show how the FBG-reﬂected
spectrum was deformed at each step of the calculation
procedures of the new method. As a ﬁrst step, we measured
the FBG-reﬂected spectrum intentionally with the lowest
resolution [2.00 nm; Fig. 4(a)]; as mentioned above, this step
is not necessary for the Lorentzian demodulation algorithm
itself, but we intended to show the feasibility of employing a
low-cost OSA with insufﬁcient resolution. Here, as an example,
the spectrum at strain of 0.5% was shown. The wavelength
range needs to be adjusted depending on the maximal strain to
be measured. As a second step, we subtracted the spectral noise
ﬂoor of the ASE from the measured spectrum [Fig. 4(b)]. As
the third step, we ﬁtted the spectrum with a Lorentzian curve[Fig. 4(c)] and derived its central wavelength as the Bragg
wavelength (1564.2 nm).
We performed these procedures to Fig. 3(a) and obtained thestrain dependence of the FBG-reﬂected spectrum, as shown in
Fig. 5(a). Note that the vertical axis was normalized so that thepeak power became 1. In the same manner as in Fig. 3(a), withincreasing strain, the Bragg wavelength shifted to a longer
wavelength. Subsequently, we plotted the Bragg wavelength asa function of strain [Fig. 5(b)]. The dependence was linear in
the strain range of up to 1.0% with no wavelength hopping, andthe calculated coefﬁcient was ∼14.1 nm/%, which is in good
agreement with the previously reported value (∼14.3 nm/%).13)

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Jpn. J. Appl. Phys. 58, 028003 (2019)

Y. Mizuno et al.

Fig. 5. (Color online) Strain dependencies of (a) the normalized FBG-reﬂected spectrum and (b) the Bragg wavelength calculated using the new method. The
dotted line is a linear ﬁt.

Note that the bandwidth of the Lorentzian curve did not exhibit
monotonic dependence on the strain [Fig. 5(a)], because eachpeak included in the spectrum shifts with its independent strain
coefﬁcient. This means that it is difﬁcult to use the Bragg
wavelength and the bandwidth simultaneously to perform
discriminative measurements of two physical parameters,such as strain and temperature.
In conclusion, we developed a simple and accurate demodulation algorithm based on the Lorentzian approximation for
detecting the Bragg wavelength of the multiple-peak FBG
spectrum. First, the maximum and centroid detection methods
were shown to lead to measurement failure, especially when
the strain was larger (>0.3%). Subsequently, the correct Bragg
wavelength detection at a strain of up to 1.0% was experimentally shown to be feasible using a PFGI-POF-FBG with
our demodulation method. We anticipate that this method will
be one of the standard tools for detecting the Bragg wavelength from a multiple-peak FBG (such as a POF-FBG)
spectrum in the future.
Acknowledgments This work was supported by JSPS KAKENHI Grant
No. 17H04930 and 17J07226 and by a research grant from the Fujikura
Foundation.
ORCID iDs Yosuke Mizuno
https://orcid.org/0000-0002-3362-4720
Heeyoung Lee
https://orcid.org/0000-0003-3179-0386
Antreas Theodosiou
https://orcid.org/0000-0002-5912-9138
Kyriacos Kalli
https://orcid.org/0000-0003-4541-092X
Kentaro Nakamura
https://orcid.org/0000-0003-2899-4484

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028003-4© 2019 The Japan Society of Applied Physics